Subgroups — C₃×#83253 #11857078

Order 1152.

1441 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#944	C₂	1	4	yes	maximal
3	C₂×C₃×#869	C₂	1	1	yes	maximal, characteristic
4	C₃×#4975	C₂	1	8	yes	maximal
5	C₃×C₄²×Dic₃	C₂	1	1	yes	maximal, characteristic
6	C₃×#2868	—	3	1	no	maximal
7	#83253	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂²×C₃²×Q₈	C₂²	1	2	yes	—
9	C₂²×C₃×Dic₆	C₂²	1	2	yes	—
10	C₃²×C₄⋊Q₈	—	2	4	no	—
11	C₂×C₃²×C₄⋊C₄	C₂²	1	4	yes	—
12	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
13	C₃×#944	—	2	16	no	—
14	C₂×C₃×C₄⋊Dic₃	C₂²	1	4	yes	—
15	C₂×C₃×Dic₃⋊C₄	C₂²	1	16	yes	—
16	C₂×C₃×C₄×Dic₃	C₂²	1	2	yes	—
17	C₂×C₃×C₄×Dic₃	C₂²	1	4	yes	—
18	C₃×#869	—	2	4	no	—
19	C₂×C₃×C₄⋊Q₈	—	2	1	no	—
20	C₂×C₃×C₄⋊Q₈	—	3	4	no	—
21	C₂×C₃×C₄⋊Q₈	—	3	1	no	—
22	C₂×C₃×C₄⋊Q₈	D₃	1	1	yes	characteristic
23	C₂×C₃×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
24	C₂×#944	C₂×C₃	1	4	yes	—
25	C₂×#869	C₂×C₃	1	1	yes	characteristic
26	C₃×#384	—	3	8	no	—
27	C₃×C₄³	—	3	1	no	—
28	#4975	C₂×C₃	1	8	yes	—
29	C₄²×Dic₃	C₂×C₃	1	1	yes	characteristic
30	C₂×C₃²×Q₈	—	2	4	no	—
31	C₂×C₃²×Q₈	—	2	8	no	—
32	C₂²×C₃²×C₄	C₂³	1	4	yes	—
33	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
34	C₂²×C₃²×C₄	C₂³	1	2	yes	—
35	C₂²×C₃×Dic₃	C₂³	1	4	yes	—
36	C₂²×C₃×Dic₃	C₂³	1	4	yes	—
37	C₂×C₃×Dic₆	—	2	8	no	—
38	C₂×C₃×Dic₆	—	2	4	no	—
39	C₃²×C₄⋊C₄	—	2	8	no	—
40	C₃²×C₄²	D₄	1	4	yes	—
41	C₃×C₄⋊Dic₃	—	2	8	no	—
42	C₃×Dic₃⋊C₄	—	2	32	no	—
43	C₃×C₄×Dic₃	D₄	1	8	yes	—
44	C₃×C₄×Dic₃	Q₈	1	16	yes	complemented
45	#2868	—	3	1	no	Sylow
46	C₂²×C₃×Q₈	—	2	2	no	—
47	C₂²×C₃×Q₈	—	3	2	no	—
48	C₂²×C₃×Q₈	C₂×D₃	1	2	yes	—
49	C₂²×C₃×Q₈	C₂²×C₃	1	2	yes	—
50	C₂²×Dic₆	C₂²×C₃	1	2	yes	—
51	C₃×C₄⋊Q₈	—	2	8	no	—
52	C₃×C₄⋊Q₈	—	2	4	no	—
53	C₃×C₄⋊Q₈	—	2	4	no	—
54	C₃×C₄⋊Q₈	—	6	16	no	—
55	C₃×C₄⋊Q₈	—	6	4	no	—
56	C₂×C₃×C₄⋊C₄	—	2	4	no	—
57	C₂×C₃×C₄⋊C₄	—	3	16	no	—
58	C₂×C₃×C₄⋊C₄	—	3	4	no	—
59	C₂×C₃×C₄⋊C₄	C₂×D₃	1	4	yes	—
60	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	4	yes	—
61	C₂×C₃×C₄²	—	2	1	no	—
62	C₂×C₃×C₄²	—	3	4	no	—
63	C₂×C₃×C₄²	—	3	2	no	—
64	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
65	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
66	#944	—	2	16	no	—
67	C₂×C₄⋊Dic₃	C₂²×C₃	1	4	yes	—
68	C₂×Dic₃⋊C₄	C₂²×C₃	1	16	yes	—
69	C₂×C₄×Dic₃	C₂²×C₃	1	2	yes	—
70	C₂×C₄×Dic₃	C₂²×C₃	1	4	yes	—
71	#869	—	2	4	no	—
72	C₂³×C₃²	C₂⁴	1	1	yes	characteristic
73	C₃²×Q₈	—	4	8	no	—
74	C₂×C₃²×C₄	—	2	4	no	—
75	C₂×C₃²×C₄	—	2	8	no	—
76	C₂×C₃²×C₄	C₂×D₄	1	4	yes	—
77	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
78	C₂×C₃²×C₄	C₂×D₄	1	4	yes	—
79	C₂×C₃²×C₄	C₂×Q₈	1	8	yes	—
80	C₂×C₃×Dic₃	—	2	8	no	—
81	C₂×C₃×Dic₃	—	2	4	no	—
82	C₂×C₃×Dic₃	C₂×D₄	1	8	yes	—
83	C₂×C₃×Dic₃	C₂×Q₈	1	16	yes	—
84	C₃×Dic₆	—	4	8	no	—
85	C₂×C₄⋊Q₈	—	3	4	no	—
86	C₂×C₄⋊Q₈	—	3	1	no	—
87	C₂×C₄⋊Q₈	C₃×D₃	1	1	yes	characteristic
88	#384	—	3	8	no	—
89	C₄³	—	3	1	no	—
90	C₂×C₃×Q₈	—	2	8	no	—
91	C₂×C₃×Q₈	—	2	8	no	—
92	C₂×C₃×Q₈	—	2	4	no	—
93	C₂×C₃×Q₈	—	2	8	no	—
94	C₂×C₃×Q₈	—	2	4	no	—
95	C₂×C₃×Q₈	—	4	8	no	—
96	C₂×C₃×Q₈	—	6	8	no	—
97	C₂×C₃×Q₈	—	6	4	no	—
98	C₂²×C₃×C₄	—	2	2	no	—
99	C₂²×C₃×C₄	—	2	1	no	—
100	C₂²×C₃×C₄	—	2	4	no	—
101	C₂²×C₃×C₄	—	3	4	no	—
102	C₂²×C₃×C₄	—	3	4	no	—
103	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
104	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
105	C₂²×C₃×C₄	C₂²×D₃	1	2	yes	—
106	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
107	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
108	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
109	C₂²×Dic₃	C₂³×C₃	1	4	yes	—
110	C₂²×Dic₃	C₂³×C₃	1	4	yes	—
111	C₂×Dic₆	—	2	4	no	—
112	C₂×Dic₆	—	2	8	no	—
113	C₃×C₄⋊C₄	—	2	16	no	—
114	C₃×C₄⋊C₄	—	2	8	no	—
115	C₃×C₄⋊C₄	—	2	8	no	—
116	C₃×C₄⋊C₄	—	6	32	no	—
117	C₃×C₄⋊C₄	—	6	8	no	—
118	C₃×C₄²	—	2	4	no	—
119	C₃×C₄²	—	3	8	no	—
120	C₃×C₄²	—	3	16	no	—
121	C₃×C₄²	#67	1	4	yes	—
122	C₃×C₄²	C₃×D₄	1	4	yes	—
123	C₄⋊Dic₃	—	2	8	no	—
124	Dic₃⋊C₄	—	2	32	no	—
125	C₄×Dic₃	C₃×D₄	1	8	yes	—
126	C₄×Dic₃	C₃×Q₈	1	16	yes	complemented
127	C₂²×C₃²	C₂²×D₄	1	2	yes	—
128	C₂²×C₃²	C₂²×D₄	1	1	yes	characteristic
129	C₂²×C₃²	C₂²×Q₈	1	4	yes	—
130	C₃²×C₄	—	4	4	no	—
131	C₃²×C₄	#127	1	4	yes	—
132	C₃²×C₄	C₄⋊Q₈	1	8	yes	complemented
133	C₃×Dic₃	—	4	4	no	—
134	C₃×Dic₃	C₄⋊Q₈	1	16	yes	complemented
135	C₂²×Q₈	—	3	2	no	—
136	C₂²×Q₈	C₂×C₃×D₃	1	2	yes	—
137	C₄⋊Q₈	—	2	4	no	—
138	C₄⋊Q₈	—	6	16	no	—
139	C₄⋊Q₈	—	6	4	no	—
140	C₂×C₄⋊C₄	—	3	16	no	—
141	C₂×C₄⋊C₄	—	3	4	no	—
142	C₂×C₄⋊C₄	C₂×C₃×D₃	1	4	yes	—
143	C₂×C₄²	—	3	2	no	—
144	C₂×C₄²	—	3	4	no	—
145	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
146	C₂³×C₃	—	2	1	no	—
147	C₂³×C₃	C₂³×D₃	1	1	yes	characteristic
148	C₂³×C₃	C₂⁴×C₃	1	1	yes	characteristic
149	C₃×Q₈	—	4	16	no	—
150	C₃×Q₈	—	4	8	no	—
151	C₃×Q₈	—	4	8	no	—
152	C₃×Q₈	—	12	8	no	—
153	C₂×C₃×C₄	—	2	8	no	—
154	C₂×C₃×C₄	—	2	8	no	—
155	C₂×C₃×C₄	—	2	4	no	—
156	C₂×C₃×C₄	—	2	8	no	—
157	C₂×C₃×C₄	—	2	4	no	—
158	C₂×C₃×C₄	—	2	4	no	—
159	C₂×C₃×C₄	—	2	8	no	—
160	C₂×C₃×C₄	—	2	4	no	—
161	C₂×C₃×C₄	—	2	2	no	—
162	C₂×C₃×C₄	—	3	8	no	—
163	C₂×C₃×C₄	—	3	16	no	—
164	C₂×C₃×C₄	—	4	8	no	—
165	C₂×C₃×C₄	—	6	8	no	—
166	C₂×C₃×C₄	—	6	4	no	—
167	C₂×C₃×C₄	D₃×D₄	1	4	yes	—
168	C₂×C₃×C₄	D₃×Q₈	1	8	yes	—
169	C₂×C₃×C₄	C₂×#67	1	2	yes	—
170	C₂×C₃×C₄	C₂×#67	1	4	yes	—
171	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
172	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
173	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
174	C₂×C₃×C₄	C₂×C₃×Q₈	1	8	yes	—
175	C₂×Dic₃	—	2	8	no	—
176	C₂×Dic₃	—	2	4	no	—
177	C₂×Dic₃	C₂×C₃×D₄	1	8	yes	—
178	C₂×Dic₃	C₂×C₃×Q₈	1	16	yes	—
179	Dic₆	—	4	8	no	—
180	C₂×C₃²	C₂×#127	1	1	yes	characteristic
181	C₂×C₃²	C₂×C₄⋊Q₈	1	4	yes	—
182	C₂×C₃²	C₂×C₄⋊Q₈	1	2	yes	—
183	C₂×Q₈	—	2	8	no	—
184	C₂×Q₈	—	2	4	no	—
185	C₂×Q₈	—	6	8	no	—
186	C₂×Q₈	—	6	4	no	—
187	C₂²×C₄	—	3	4	no	—
188	C₂²×C₄	—	3	4	no	—
189	C₂²×C₄	C₂²×C₃×D₃	1	4	yes	—
190	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
191	C₂²×C₄	C₂²×C₃×D₃	1	2	yes	—
192	C₄⋊C₄	—	2	8	no	—
193	C₄⋊C₄	—	6	32	no	—
194	C₄⋊C₄	—	6	8	no	—
195	C₄²	—	3	8	no	—
196	C₄²	—	3	16	no	—
197	C₄²	C₃×#67	1	4	yes	—
198	C₂²×C₃	—	2	2	no	—
199	C₂²×C₃	—	2	4	no	—
200	C₂²×C₃	—	2	1	no	—
201	C₂²×C₃	C₂×D₃×D₄	1	2	yes	—
202	C₂²×C₃	C₂×D₃×Q₈	1	4	yes	—
203	C₂²×C₃	C₂²×#67	1	1	yes	characteristic
204	C₂²×C₃	C₂²×C₃×D₄	1	2	yes	—
205	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
206	C₂²×C₃	C₂²×C₃×Q₈	1	4	yes	—
207	C₃×C₄	—	2	4	no	—
208	C₃×C₄	—	2	8	no	—
209	C₃×C₄	—	3	16	no	—
210	C₃×C₄	—	4	8	no	—
211	C₃×C₄	—	4	4	no	—
212	C₃×C₄	—	4	4	no	—
213	C₃×C₄	—	12	4	no	—
214	C₃×C₄	#940	1	4	yes	—
215	C₃×C₄	#944	1	8	yes	complemented
216	C₃×C₄	C₃×#127	1	4	yes	—
217	C₃×C₄	C₃×C₄⋊Q₈	1	8	yes	complemented
218	Dic₃	—	4	4	no	—
219	Dic₃	C₃×C₄⋊Q₈	1	16	yes	complemented
220	C₃²	#2868	1	1	yes	characteristic, complemented, Sylow
221	C₂³	C₂³×C₃×D₃	1	1	yes	characteristic
222	Q₈	—	4	8	no	—
223	Q₈	—	12	8	no	—
224	C₂×C₄	—	2	8	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	3	8	no	—
227	C₂×C₄	—	3	16	no	—
228	C₂×C₄	—	6	8	no	—
229	C₂×C₄	—	6	4	no	—
230	C₂×C₄	C₃×D₃×D₄	1	4	yes	—
231	C₂×C₄	C₃×D₃×Q₈	1	8	yes	—
232	C₂×C₄	C₂×C₃×#67	1	2	yes	—
233	C₂×C₄	C₂×C₃×#67	1	4	yes	—
234	C₂×C₃	—	2	1	no	—
235	C₂×C₃	—	2	2	no	—
236	C₂×C₃	—	2	4	no	—
237	C₂×C₃	D₃×C₄⋊Q₈	1	4	yes	—
238	C₂×C₃	C₂×#940	1	1	yes	characteristic
239	C₂×C₃	C₂×#944	1	2	yes	—
240	C₂×C₃	C₂×C₃×#127	1	1	yes	characteristic
241	C₂×C₃	C₂×C₃×C₄⋊Q₈	1	2	yes	—
242	C₂×C₃	C₂×C₃×C₄⋊Q₈	1	4	yes	—
243	C₂²	C₂×C₃×D₃×D₄	1	2	yes	—
244	C₂²	C₂×C₃×D₃×Q₈	1	4	yes	—
245	C₂²	C₂²×C₃×#67	1	1	yes	characteristic
246	C₄	—	3	16	no	—
247	C₄	—	4	4	no	—
248	C₄	—	12	4	no	—
249	C₄	C₃×#940	1	4	yes	—
250	C₄	C₃×#944	1	8	yes	complemented
251	C₃	—	2	1	no	—
252	C₃	#83253	1	1	yes	characteristic, direct product factor
253	C₃	C₃×#2868	1	1	yes	characteristic, complemented
254	C₂	C₃×D₃×C₄⋊Q₈	1	4	yes	—
255	C₂	C₂×C₃×#940	1	1	yes	characteristic
256	C₂	C₂×C₃×#944	1	2	yes	—

	subgroup	G/H
center	C₂³×C₃	C₂³×D₃
commutator, cosocle	C₂³×C₃	C₂⁴×C₃
Frattini	C₂³	C₂³×C₃×D₃
Fitting	C₂×C₃²×C₄⋊Q₈	C₂
socle, prime radical	C₂³×C₃²	C₂⁴
hypercenter	C₂×C₃×C₄⋊Q₈	D₃
hypocenter	C₃	C₃×#2868
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄⋊Q₈

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
7. #83253	C₃	direct (N × H)
44. C₃×C₄×Dic₃	Q₈	semidirect
126. C₄×Dic₃	C₃×Q₈	semidirect
132. C₃²×C₄	C₄⋊Q₈	semidirect
134. C₃×Dic₃	C₄⋊Q₈	semidirect
215. C₃×C₄	#944	semidirect
217. C₃×C₄	C₃×C₄⋊Q₈	semidirect
219. Dic₃	C₃×C₄⋊Q₈	semidirect
220. C₃²	#2868	semidirect
250. C₄	C₃×#944	semidirect
252. C₃	#83253	direct (N × H)
253. C₃	C₃×#2868	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃²×C₄⋊Q₈	2
C₂×C₃×#944	2
C₂×C₃×#869	2
C₃×#4975	2
C₃×C₄²×Dic₃	2
C₃×#2868	3
#83253	3

Subgroups — C3×#83253 #11857078

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#83253 #11857078